The system under interest is studied in the context of non-equilibrium thermodynamics [1]}, [2]}, [3]}, [4]}, [5]}, [6]}. It is a thin homogeneous conducting layer of length \(L\)  and width \(\ell \)  contacted to an electric generator, and submitted to a constant magnetic field \(\vec{H}\)  oriented along the \(z\)  axis (see Fig.REF ).
We assume that the conducting layer is planar, invariant by translation along the \(x\)  axis \(\ell \ll L\)  (this excludes the region in contact with the power generator), and the two lateral edges are symmetric.
